As a follow-up to this earlier question on Fraenkel and Klein, I am interested in the actual contents of the article Fraenkel apparently wrote for Klein's volume on Gauss, which I don't have access to. Did Fraenkel's article actually discuss set theory which was Fraenkel's specialty?
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$\begingroup$ Here you find your grandpa's original paper: gdz.sub.uni-goettingen.de/dms/load/img/… But I am too lazy to check the 240 pages for "Mengenlehre". $\endgroup$– user5707May 5, 2017 at 19:02
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$\begingroup$ @user37237, Abraham Adolf Fraenkel is not my grandpa but rather the grandpa of another editor who recently turned up at HSM (this explains my reference to grandpere in an earlier question). $\endgroup$– Mikhail KatzMay 7, 2017 at 12:01
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Fraenkel's article does not discuss set theory but gives a review, based on Gauss' letters and notes, how Gauss understood and defined the numbers in comparison with Bolzano, Cauchy, Dedekind, and others.
Only on p. 5 Fraenkel asserts that Gauss' famous quote "I protest firstly against the use of an infinite magnitude as a completed one, which never has been allowed in mathematics" does not mean what it appears to mean to the avearge reader and does not condemn the infinite in general. He claims that Gauss would have agreed with Cantor.
On the last pages 48-49, Fraenkel mentiones sytems with infinitely many dimensions, but not set theory.
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$\begingroup$ Thanks very much! This is tremendous. I am very interested in how Fraenkel understood how Gauss, Bolzano, and Cauchy defined numbers. For example, Cauchy does not say anything about this himself. Is there by any chance a translation of this article by Fraenkel? If not, I would much appreciate if you could summarize the main points. $\endgroup$ May 7, 2017 at 12:00
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$\begingroup$ The text is not very exciting. Here are some points: Like Bolzano Gauss takes the existence of a real root of every algebraic equation of odd degree as given (constat). Before Cauchy Gauss had developed the notion of limit of a sequence of numbers. So he was the first to complete the project, started by Wallis, of a purely arithmetical definition of the notion of limit. The sum of converging sequences converges to the sum of the limits. No reference is given to the product because Gauss' corresponding notes were interrupted, probably by his Disquisitiones and remained fragmentary. $\endgroup$– user5737May 7, 2017 at 18:16
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$\begingroup$ The crucial idea of creating new numbers by sequences (or sets) of known numbers is not detectable. Dirichlet seems to have been the first who points to that topic, as noted by Dedekind. Gauss seems to have overlooked that a two dimensional grid, a sequence of sequences, can be ordered as a single sequence (compare aleph_0 * aleph_0 = aleph_0). And so on. Not very exciting. $\endgroup$– user5737May 7, 2017 at 18:17